a. Suppose that
+ X2, where
are independent Cauchy random variables, each having pdf
Using the integral identity
S2 has a pdf 2/[(4 + y2)].
b. Proceeding by induction, show that
+ X2 + . . . +
has a pdf n / [(n2 + Y2)].
c. Thus, verify that the average of n
independent Cauchy samples
Sn / n) has a
Cauchy pdf 1 / [(n2 + Y2
"averaging together" a number of independent Cauchy
samples yields a pdf for the
average identical to that of any one of the individual samples.
(This result contrasts
sharply to most random variables, for which averaging of n
independent samples reduces the
variance by a factor of n^-1.)